You can figure out the differentials [itex]dx[/itex] and [itex]dy[/itex] from the general formula for a multivariable differential: $$ x = x(r, \theta) = r \cos \theta \Leftrightarrow dx = \frac{\partial x}{\partial r}dr+\frac{\partial x}{\partial \theta}d\theta = \cos \theta\,dr - r \sin \theta\,d\theta $$
$$ y = y(r, \theta) = r \sin \theta \Leftrightarrow dy = \frac{\partial y}{\partial r}dr+\frac{\partial y}{\partial \theta}d\theta = \sin \theta\,dr + r \cos \theta\,d\theta $$
For the relation between the area elements [itex]dA[/itex], you have to use the Jacobian:
$$ dx dy = \frac{\partial(x,y)}{\partial(r,\theta)}dr\,d\theta = (r \cos^2 \theta + r \sin^2 \theta)\,dr\,d\theta = r\,dr\,d\theta
$$
This is also easily seen if you sketch out what is happening. If you change [itex]\theta[/itex] by a small amount, you change one side of your area element by [itex]r\,d\theta[/itex]. If you change [itex]r[/itex] by a small amount, you change the other side of your area element by [itex]dr[/itex].

As you can see, JPaquim's formula for the Jacobian is simply the determinant of the transformation matrix between the differentials.
And if you remember your linear algebra, it is precisely the determinant of a square matrix that tells you of how the area is transformed when going from one set of basis vectors to another by means of matrix transformation.